TY - GEN A1 - Borndörfer, Ralf A1 - Schwartz, Stephan A1 - Surau, William T1 - Rooted Maximum Weight Connected Subgraphs with Balancing and Capacity Constraints T3 - ZIB-Report - 21-34 Y1 - 2021 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-84427 SN - 1438-0064 ER - TY - CHAP A1 - Borndörfer, Ralf A1 - Schwartz, Stephan A1 - Surau, William T1 - Finding Minimum Balanced Separators - an Exact Approach T2 - Operations Research Proceedings 2021 N2 - Balanced separators are node sets that split the graph into size bounded components. They find applications in different theoretical and practical problems. In this paper we discuss how to find a minimum set of balanced separators in node weighted graphs. Our contribution is a new and exact algorithm that solves Minimum Balanced Separators by a sequence of Hitting Set problems. The only other exact method appears to be a mixed-integer program (MIP) for the edge weighted case. We adapt this model to node weighted graphs and compare it to our approach on a set of instances, resembling transit networks. It shows that our algorithm is far superior on almost all test instances. T3 - ZIB-Report - 21-25 Y1 - 2021 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-83138 SN - 1438-0064 SP - 154 EP - 159 ER - TY - GEN A1 - Borndörfer, Ralf A1 - Schwartz, Stephan A1 - Surau, William T1 - Vertex Covering with Capacitated Trees N2 - The covering of a graph with (possibly disjoint) connected subgraphs is a fundamental problem in graph theory. In this paper, we study a version to cover a graph's vertices by connected subgraphs subject to lower and upper weight bounds, and propose a column generation approach to dynamically generate feasible and promising subgraphs. Our focus is on the solution of the pricing problem which turns out to be a variant of the NP-hard Maximum Weight Connected Subgraph Problem. We compare different formulations to handle connectivity, and find that a single-commodity flow formulation performs best. This is notable since the respective literature seems to have dismissed this formulation. We improve it to a new coarse-to-fine flow formulation that is theoretically and computationally superior, especially for large instances with many vertices of degree 2 like highway networks, where it provides a speed-up factor of 10 over the non-flow-based formulations. We also propose a preprocessing method that exploits a median property of weight constrained subgraphs, a primal heuristic, and a local search heuristic. In an extensive computational study we evaluate the presented connectivity formulations on different classes of instances, and demonstrate the effectiveness of the proposed enhancements. Their speed-ups essentially multiply to an overall factor of 20. Overall, our approach allows the reliabe solution of instances with several hundreds of nodes in a few minutes. These findings are further corroborated in a comparison to existing districting models on a set of test instances from the literature. T3 - ZIB-Report - 21-14 Y1 - 2021 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-82616 SN - 1438-0064 ER - TY - JOUR A1 - Borndörfer, Ralf A1 - Schwartz, Stephan A1 - Surau, William T1 - Vertex Covering with Capacitated Trees JF - Networks N2 - The covering of a graph with (possibly disjoint) connected subgraphs is a funda-mental problem in graph theory. In this paper, we study a version to cover a graph’svertices by connected subgraphs subject to lower and upper weight bounds, and pro-pose a column generation approach to dynamically generate feasible and promisingsubgraphs. Our focus is on the solution of the pricing problem which turns out to bea variant of the NP-hard Maximum Weight Connected Subgraph Problem. We com-pare different formulations to handle connectivity, and find that a single-commodityflow formulation performs best. This is notable since the respective literature seemsto have widely dismissed this formulation. We improve it to a new coarse-to-fine flowformulation that is theoretically and computationally superior, especially for largeinstances with many vertices of degree 2 like highway networks, where it provides aspeed-up factor of 5 over the non-flow-based formulations. We also propose a pre-processing method that exploits a median property of weight-constrained subgraphs,a primal heuristic, and a local search heuristic. In an extensive computational studywe evaluate the presented connectivity formulations on different classes of instances,and demonstrate the effectiveness of the proposed enhancements. Their speed-upsessentially multiply to an overall factor of well over 10. Overall, our approach allowsthe reliable solution of instances with several hundreds of vertices in a few min-utes. These findings are further corroborated in a comparison to existing districtingmodels on a set of test instances from the literature Y1 - 2023 U6 - https://doi.org/10.1002/net.22130 VL - 81 IS - 2 SP - 253 EP - 277 ER -